Papers

We study exact fillings of Legendrians and contact manifolds using Floer theory of universal covers. We prove vanishing results on wrapped Floer homology of the universal cover and symplectic (co)homology with $k[\pi_1]$ coefficients by using lifted versions of Ritter's TQFT operations. We use these results to show that an exact Maslov zero Lagrangian filling of the standard Legendrian $\Lambda_{std}$ in any Liouville filling of $(S^{2n-1},\xi_{std})$ is diffeomorphic to a ball in dimension at least 5. We show that the existence of a topological simple exact filling $W$ with vanishing symplectic homology for a dynamically convex contact manifold $M^{2n-1}$ reveals information about homotopy groups of $W$ and $M.$ We also strengthen a theorem of Zhou for ADC-contact manifolds as an application of universal covers.

We study bijective nonexpansive maps in $T_0$-quasi-metric spaces, distinguishing preservation of directed distances from pointwise rigidity. Our focus is their behaviour under passage to the $q$-hyperconvex hull. The upper quasi-metric integers are $q$-plastic and bicomplete, but their hull is the non-$q$-plastic upper real line. An explicit hull calculation and an established metric dense-rigidity theorem give the same failure of inheritance for a $q$-rigid original space. Tightness propagates distance preservation from the canonical copy throughout the hull. Under compactness and a bounded lattice structure for the zero-distance order, identifying that copy with the complemented elements gives a sufficient condition for inheritance of $q$-rigidity. A four-vertex asymmetric rectangle verifies this criterion. We also record metric transfer mechanisms and show that a directed interval condition gives an endpoint-coordinate representation, automatic $q$-plasticity, and $q$-rigidity under uniqueness of the ordered diametral pair.

A negative answer to Whitehead's asphericity problem would follow from an infinite chain of two-complexes, starting with a non-aspherical one, in which every inclusion induces the zero map on second homotopy groups. We prove that a connected two-complex $K$ is the first term of such chains of every finite length if and only if $K$ has a connected acyclic regular cover. In particular, there is a finite non-aspherical two-complex $K$, a presentation complex of $\mathrm{SL}(2,5)$, such that for every $n$ there is a strictly increasing chain of finite two-complexes $K=K_0\subset K_1\subset\dots\subset K_n$ in which all inclusions are zero on $\pi_2$.

Let $H_n$ be the one-dimensional Honda formal group of height $n$ over $\mathbf F_{p^n}$ and let \[ R_n=W(\mathbf F_{p^n})[[u_1,\ldots,u_{n-1}]],\qquad A_n=R_n/(p). \] We prove, for every height $n$ and every prime $p$, that the prime ideals of $A_n$ stable under an open subgroup of the Morava stabilizer group are exactly the height ideals $(u_1,\ldots,u_j)$. We also prove that a stable prime of $R_n$ avoiding $p$ is zero. The resulting radical-ideal classification implies the Hovey-Strickland classification of thick tensor ideals in the category of dualizable $K(n)$-local spectra via the forward implication of Barthel-Heard-Naumann. The special-fiber argument is local and geometric. After cutting an invariant prime by a one-parameter curve, we construct from the Cartier structure equation a smooth formal quotient $\mathcal{Q}$ and a distinguished subgroup $\mathcal{H}$. The generic fiber of $\mathcal{H}$ is identified with the deformation space of connected-étale extensions. A Cartier obstruction map from the full Honda endomorphism order is compared with evaluation on Tate vectors through completed universal covers. Fargues-Fontaine vector bundles give a period-detection statement. Chai's rigidity theorem then promotes detection by homomorphisms to formal Zariski density after a renormalization of the valuation. A uniform fixed-jet argument transfers this density to Morava-stabilizer orbits. The generic-fiber assertion is proved separately from the Gross-Hopkins period map.

Farrell and Jones showed that negatively curved manifolds in dimension $\geq 5$ are topologically rigid in the sense that homotopy equivalence implies homeomorphism. In the smooth category, the result does not hold even for hyperbolic manifolds, and there are negatively curved manifolds $N$ homotopy equivalent to a given hyperbolic manifold $M$ but not diffeomorphic. For complex hyperbolic manifolds, the analogous construction is demonstrated only in dimensions of the form $8n+2$. In this paper, we prove this result in many other dimensions. The main approach involves the use of $EO$-theory of complex projective spaces to construct suitable examples of exotic spheres. These theories may be viewed as a version of real $K$-theory, and are defined at each prime $p$. The computation also yields positive results about the existence of free smooth $S^1$ and $S^3$ actions on exotic spheres which do not bound parallelizable manifolds.

For $r\ge1$, let $K=2r+14$ and $d=\dim S_K$. We prove strict positivity for determinants of Mellin period functionals, yielding a full-spark theorem for periods in a fundamental half-range and, in particular, the mixed odd–even period independence conjecture of Xue. As a consequence, the consecutive first Rankin–Cohen brackets $[E_{2r+10-2j},E_{2j+2}]_1$, $1\le j\le d$, form a basis of $S_K$, and we determine the signs of all admissible ordered determinants of first Eisenstein brackets. This gives a uniform exact all-weight formula for the divisor–tau convolution $C_r(n)=\sum_{m=1}^{n-1}\sigma_{2r+1}(m)\tau(n-m)$. Reducing the same coordinate identity modulo primes, we obtain canonical prime-wise reductions for infinitely many primes in every weight and characterize sparse Ramanujan-type specializations through vanishing Cramer coordinates. Finally, for homogeneous $f,g\in\mathbf Q[E_4,E_6]$, we prove $[f,g]_1/\Delta=-3456\det((f_{E_4},f_{E_6}),(g_{E_4},g_{E_6}))$, reducing the Cramer system to a one-variable coordinate problem in $T=E_6^2/E_4^3$.

We establish an almost-splitting theorem for spacetimes under timelike sectional curvature bounds in the spirit of the celebrated Riemannian result due to Cheeger and Colding. More concretely, we show that if a globally hyperbolic spacetime with timelike sectional curvature bounded below by a nonpositive constant close to zero contains a long maximizing timelike segment, then causal diamonds along that segment are close in the Lorentzian Gromov–Hausdorff sense of Minguzzi–Suhr to those in a product metric spacetime. Our result can be interpreted as a quantitative analogue of the Lorentzian splitting theorem for spacetimes with nonnegative timelike sectional curvature by Beem–Ehrlich–Markvorsen–Galloway.

This paper concerns the planar Liénard system \(\dot x=y-F(x),\ \dot y=-x\), where \(F(x)\) is a continuous piecewise linear function with exactly \(n\) fold points. Tonnelier [SIAM J. Appl. Math. 63 (2002)] conjectured that the maximum number of limit cycles of the system is \(n\) when \(F\) has exactly \(n\) fold points. The conjecture was confirmed for \(n=1\) and \(n=2\) in [J. Nonlinear Sci. 25 (2015)] and [J. Lond. Math. Soc. 113 (2026)], respectively, whereas the case \(n\ge3\) remained open. In this paper, we show that the maximal number of limit cycles is at least \(|3n-4|\) for \(n\in\mathbb N^+\), thereby disproving Tonnelier's conjecture for \(n\ge3\). The proof reveals a unified multiscale perturbation mechanism underlying the creation and coexistence of multiple limit cycles. Moreover, we prove that the number of limit cycles admits the finite upper bound \(2^{28(n+1)^2}\).

On-shell forms are differential forms on the Grassmannian which arise in particle physics. They are defined using bipartite graphs with $n$ distinguished “boundary” vertices. Mathematical investigation of on-shell forms has largely focused on the case where the graph is planar, in which case one can utilize combinatorial tools pioneered by Postnikov in the study of the totally nonnegative Grassmannian. In this article, we investigate on-shell forms for arbitrary graphs. We first discuss how to extend various tools from the planar case to arbitrary graphs. We then prove a determinantal formula for a class of on-shell forms which first appeared in physics literature. We explain the relation between this class of forms and the hypertree divisors of $M_{0,n}$, introduced by Castravet–Tevelev.

We study the tableaux monoid from the perspective of rewriting theory. We first construct a finite quadratic convergent presentation based on column generators and prove that it yields a biautomatic structure. We then extend this presentation to a finite coherent presentation in the sense of polygraphic rewriting. These results provide canonical normal forms, yield homological and algorithmic consequences, and produce a higher-dimensional presentation that can be used to describe actions of the tableaux monoid on categories.

Ferromagnetic spin models on random regular graphs, such as the Ising and Potts models, have been studied extensively. Several equivalent variational and recursive representations of their limiting pressure are known, and these representations can be used to establish agreement between the quenched and annealed pressures. We give a concise overview of these representations and highlight the connections among the different formulations. We then derive analogous representations for the exponential growth rate of the second moment of the partition function, and prove that it has the same order as the square of the annealed pressure, up to a subexponential error. Combined with concentration of the quenched pressure, this identity yields a transparent second-moment proof of quenched–annealed agreement on random regular graphs. By bringing these representations and arguments together in a unified presentation, we aim to provide an accessible entry point to the literature on ferromagnetic spin models on random regular graphs.

A theoretical understanding of multi-head self-attention is fundamental to the study of modern neural networks. Using random matrix theory, we establish Gaussian equivalence for multi-head self-attention: replacing softmax attention with rescaled scores plus Gaussian noise preserves the limiting spectral law of the centered output. This equivalence also covers value and output projections that depend on the keys. The resulting laws separate the effects of head allocation and projection widths, and distinguish spectrum-preserving across-head sharing from within-head key–value dependence.

Massive multiple-input multiple-output (MIMO) achieves a high spectral efficiency (SE) by employing a large number of receive antennas in the uplink. Employing many receive antennas requires a large number of analog-to-digital converters (ADCs) and a high fronthaul data rate. Low-resolution ADCs can reduce power consumption and cost, while fronthaul quantization can reduce the required fronthaul data rate, at the expense of quantization distortion at both stages. In wideband orthogonal frequency-division multiplexing systems, ADC quantization is performed in the time domain, while fronthaul quantization may be applied after the discrete Fourier transform to transmit only active subcarriers. In this study, we investigate the achievable uplink SE over fronthaul links under double quantization and imperfect channel estimation. We derive a linear minimum mean-squared error channel estimator from the double-quantized pilot observations and obtain an achievable SE using the use-and-then-forget bound. Numerical results demonstrate that the ADC and fronthaul quantization resolutions have comparable impacts on the achievable SE. When the two resolutions differ, the lower resolution becomes the dominant factor limiting the SE. Furthermore, our numerical results show that six-bit ADC and fronthaul quantization achieve more than 90% of the SE attained in the ideal case.

We study the new eigenvalues of a fixed point interaction on the cubic three-torus. Their mean-one consecutive spacings converge to a probability law independent of the interaction parameter. We characterize this law by the consecutive zeros of a random meromorphic function built from three-squares congruence densities. Its small-gap distribution has the form $s^5\Psi(\log_2(s^{-2}))+o(s^5)$, where $\Psi$ is continuous, positive, and one-periodic. The proof combines periodic mean-square approximation, a weighted Hilbert-transform estimate, and Fourier estimates that retain the interactions between primes.

We study a nonlocal stochastic optimal control problem in which the control acts through the elliptic smoothing operator $\mathcal{S}=(I-\Delta)^{-1}$ on $\mathbb{R}^d$. The state is described by the law of a controlled diffusion, equivalently by a controlled Fokker-Planck equation with coefficients depending on $\mathcal{S}u$. We prove the existence of an optimal control by the direct method in the calculus of variations. We then derive a Pontryagin-type maximum principle by spike variations, relying on sharp properties of the operator $\mathcal{S}$. The result yields a pointwise minimisation rule for the optimal control. We discuss the meaning of the optimal control problem in the case of population dynamics.

We prove a prime number theorem for fixed terminal block words in Pisot integer numeration systems. If the dominant root $\varphi$ is a Pisot number and the characteristic polynomial $P_h$ is its minimal polynomial, every terminal block word of total digit-length $m$ occurs among the primes with asymptotic frequency $\varphi^{-m}$. In the Zeckendorf case this resolves a recent conjecture. The proof converts terminal conditions into Rauzy cylinder windows and then into a linear orbit on a compact torus. For these companion substitutions, the required multiplicity-one Rauzy geometry is automatic: Barge's pure-discreteness theorem applies after reversal of the substitution words. The Rauzy torus also gives a prime number theorem for the substitution fixed word: every finite factor occurs at prime starting positions with its ordinary factor frequency. This proves the Tribonacci prime-number theorem suggested by Drmota–Müllner–Spiegelhofer. The toral model further yields polynomial sampling laws, asymptotic independence from fixed congruence classes, fixed-shift correlation formulas, and Möbius orthogonality. Combined with established prime theorems, it gives a terminal refinement of Chebotarev and shows that every fixed terminal prime class contains arbitrarily long arithmetic progressions with polylogarithmically bounded common difference.

In this paper, we establish two nonasymptotic Berry–Esseen bounds over convex sets for the Gaussian approximation of multivariate nonlinear statistics. The statistics of interest can be written as a sum of independent centered random vectors plus a remainder that may depend on all observations. The first bound retains the classical factor $d^{1/4}$ in the contribution of the independent sum, where $d$ is the dimension, while controlling the remainder through its size and its sensitivity to replacing a single observation. The second bound expresses the contribution of the independent sum in terms of fourth moments and can allow the dimension to grow faster with the sample size. For sums of independent random vectors, it removes the logarithmic factor from an existing fourth moment bound without imposing additional moment assumptions. As applications, we apply these results to Polyak–Ruppert averaging for nonsmooth stochastic approximation, temporal difference learning with linear function approximation, and multivariate $U$-statistics. The resulting bounds provide explicit Gaussian approximation errors and sufficient conditions under which these errors converge to zero as the dimension grows with the sample size.

We show that a knot in the 3-sphere with a nontrivial Jones polynomial of span at most 14 admits no purely cosmetic surgery. This extends a result of the first author for span at most 11. We determine all Laurent polynomials of span 15 satisfying the conditions on the Jones polynomial used in the proof. These polynomials form an infinite family. We also show that a knot of braid index at most 4 whose Jones polynomial has odd span admits no purely cosmetic surgery.

We study continuous-time quantum walks on connected strongly regular graphs that are not complete, observed at a random time drawn from a freely chosen probability law. Uniform average mixing (UAM) asks for a law under which every averaged transition probability equals $1/n$, where $n$ is the number of vertices. On a strongly regular graph this is equivalent to two affine constraints on three cosine moments. We construct a bounded, compactly supported time density for every strongly regular graph with nonintegral eigenvalues. For integral spectra we give an exact finite Toeplitz criterion and its Hankel form. Every averaged mixing matrix of such a graph is realized by at most two observation times. Three elementary inequalities on the moment line, which also give a short proof of Chan's classification of complex Hadamard matrices in the Bose-Mesner algebra, lead to a determination of all strongly regular graphs that admit UAM. Apart from the conference graphs of nonsquare order and the graphs with instantaneous uniform mixing, these are the members of two infinite families of parameter sets and their complements, and for them we give explicit laws with two observation times. The Petersen graph and its complement are the smallest members. A strongly regular graph with UAM admits a bounded time density exactly when it has no instantaneous uniform mixing. We also correct the classification of instantaneous uniform mixing on strongly regular graphs by Godsil, Mullin and Roy. Its sign condition excludes the halved $5$-cube, which mixes uniformly at time $\pi/4$. With the order $4\theta^2$ read literally, its parity condition also excludes the Clebsch graph and includes the parameters $(36,14,4,6)$, for which no time law gives uniform average mixing.

We consider a thermodynamically consistent diffuse-interface model for binary mixtures of incompressible viscous Newtonian fluids with a soluble surfactant. This hydrodynamic system generalizes the Abels-Garcke-Grün model for two-phase flows with unmatched densities by incorporating an additional Cahn-Hilliard equation for the surfactant concentration. We study the strong well-posedness of the initial-boundary value problem with physically relevant singular potentials, subject to a no-slip boundary condition for the fluid velocity and homogeneous Neumann boundary conditions for the phase-field variables and the corresponding chemical potentials. First, we establish the existence of local-in-time strong solutions in three dimensions and global-in-time strong solutions in two dimensions. In the two-dimensional case, we further show that, under natural growth assumptions on the singular potentials, strong solutions remain strictly separated from the pure phases, thereby guaranteeing uniqueness. In the three-dimensional case, we establish local-in-time uniqueness of strong solutions, provided that the initial data are strictly separated from the pure phases. Our results extend previous work on strong solutions to the Abels-Garcke-Grün model from the special case of constant mobility to the more general scenario with concentration-dependent mobilities and interactions with a surfactant.

In this paper, we investigate the stability threshold of the two-dimensional Boussinesq system around the Couette flow in an infinite channel $\mathbb{R}\times[-1,1]$ under Navier-slip boundary condition. To address nonlinear echo chains induced by temperature-vorticity coupling, we construct a two-level frequency-dependent temporal decomposition, with the two levels matched to the infinite and finite superposition principles for the equation, respectively. By fully exploiting the enhanced dissipation effect at each decomposition level, we prove that if the initial perturbation $(\omega^{in}, \theta^{in})$ around the Couette flow satisfies $\|\omega^{in}\|_{H^4_{x,y}\cap L^1_xH^4_y}\leq c\nu^{\frac{1}{3}}$, and $\|\theta^{in}\|_{H^5_{x,y}\cap L^1_xH^5_y}\leq c\nu^{\frac{2}{3}+}$, the Boussinesq system admits a globally asymptotically stable solution. It should be emphasized that the method in this paper provides an effective approach for the hydrodynamic equations with coupling effects.

We study the randomness required to share possibly correlated secret bits among parties connected by a graph. A dealer places shares on the edges so that each party can recover its own secret from its incident shares and learn nothing about the others beyond what its own secret reveals. Anilkumar et al. completely determined the minimum randomness required for three binary secrets. We extend this study to four secrets and obtain results for arbitrary numbers of secrets on general graphs. For four parties on a complete graph, we determine the minimum number of random states for every set of permitted secret combinations: the possible values are one, two, three and four. We also characterize when one random bit suffices on an arbitrary graph.

This paper addresses the challenge of encoding biochemical GC-content constraints in composite DNA-based data storage. Previous deterministic models impose high rate penalties by avoiding any possibility for a strand to fall outside the allowed range. To account for the stochastic nature of composite DNA, we introduce an $\epsilon$-probabilistic constraint framework, and derive capacity bounds for global constraints using composition types and for local sliding-window constraints via finite-state Markov chains. Furthermore, we propose a capacity-achieving multi-type enumerative encoder.

The ionization conjecture for the Thomas-Fermi-Dirac-von Weizsäcker model asks how many electrons a given nuclear configuration can bind. So far, only the single-center case has been settled by Frank, Nam, and Van Den Bosch, while the original molecular case has still remained open. We advance this conjecture in the two-center case. We prove that for two distinct nuclei of total charge $Z$, the number $N$ of bound electrons satisfies $N\le Z+C Z^{153/154}(1+\log Z)^{3/8}$, with constants independent of the nuclear positions and the nuclear charges. The proof relies on a comparison with the neutral Thomas-Fermi density and on localization and exterior $L^1$-estimates coupled through the screened potential. Our results could be extended to molecules with any fixed number $K$ of nuclei.

The proposed approach relies on high-frequency multivariate time series collected at the Seine aval (SIAAP 1 ) treatment plant, including pH, temperature, conductivity, and total suspended solids, as well as exogenous variables related to precipitation and measurements from an upstream plant. The objective is to forecast wastewater quality over a 24-hour horizon. A multivariate Long Short-Term Memory (LSTM) recurrent neural network is implemented to capture complex temporal dependencies and nonlinear patterns in the data. The model is trained on one year of data and validated on four months of data (one per season), and is compared with a persistence model and SARIMA models. The evaluation shows an overall superiority of the LSTM model, particularly for variables exhibiting high levels of noise and nonlinearity.

We define coarse variants of the notions of connectivity, hamiltonicity, toughness, and spanning trees. Our notions are invariant under quasi-isometries, and they obey the same relationships between them as their classical analogs. We pose several open problems that are coarse versions of classical results and conjectures about hamiltonicity.

Mathematical programming formulations are routinely modified through algebraic reformulations, perturbations of constraint functions, and the addition or removal of constraints. Some of these modifications preserve the feasible set, whereas others reflect newly imposed modeling requirements and therefore change the optimization problem itself. This paper develops a metric framework for quantifying the robustness of formulation-dependent properties. Formulations are modeled as typed collections of constraint functions. Within each fixed typed-cardinality class, their distance is defined by optimally matching constraints of the same type and measuring the resulting discrepancies in a norm on the underlying function space. As a concrete demonstration of the framework, we study the Linear Independence Constraint Qualification (LICQ). We introduce the pointwise LICQ radius, defined as the distance from a feasible formulation to the set of formulations that remain feasible at a prescribed point but fail LICQ there. Under suitable assumptions, we derive an explicit formula for this radius and characterize the effect of adding a single equality or inequality constraint.

Given a convex polytope $P\subset\Bbb R^d$ with $\mathcal F(P)$ many flags, we prove $$\operatorname{vol}(P) \operatorname{vol}(P-P)^\circ \le \frac{\mathcal F(P)}{(d!)^2}.$$ This implies the following relation between Mahler volume and the number of flag conjectured by Freij, Schmitt, Schymura and Ziegler: for a centrally symmetric polytope $P\subset\Bbb R^d$ holds $$\operatorname{vol}(P)\operatorname{vol}(P^\circ) \le \frac{2^d}{(d!)^2} \mathcal F(P).$$ This shows that the Mahler conjecture implies Kalai's flag conjecture.

We prove the finiteness of the integral of the top power of the Ricci form on a complete, noncompact Kähler manifold with positive bisectional curvature. This settles a special case of a conjecture of Yau and Yang. An immediate corollary is the finiteness of the fundamental group of such manifolds. As in our previous work, the main idea involves the construction and use of plurisubharmonic weight functions with finite Monge-Ampère masses.

We study adaptive stabilization of known control-affine systems by treating the current parameterized critic itself as a Lyapunov candidate. Its total derivative contains a parameter-velocity term that can compensate unfavorable state evolution. A projection onto the intersection of a bounded admissible velocity set and one decay halfspace enforces parameter invariance and Lyapunov decrease simultaneously. We give an exact pointwise feasibility test, a conditional state-stability theorem, and a counterexample showing why a positive critic family containing a stabilizing member does not imply persistent feasibility. A convex endpoint constraint provides a distinct sampled implementation. A generic periodic polynomial-factor critic with a local Riccati anchor is evaluated on a force-limited cart-pole without mechanical-energy features, replay training, waypoints, or controller switching. The frozen controller passes stabilization and numerical decay checks on 600 sampled initial states across two draw sets; 400 matched continuous trajectories also pass. Independent reintegration audits and a conservative local feasibility certificate are distinguished from a global or machine-arithmetic guarantee. Compared with nonlinear model-predictive control, the measured implementation uses less CPU but has higher cost, settling time, and cart travel. The results support a conditional adaptive stabilization mechanism.